确定$L$ -函数族平均秩界的最优检验函数
Determining Optimal Test Functions for Bounding the Average Rank in Families of $L$-Functions
摘要：
给定一个$L$ -函数，最重要的问题之一是它在中心点的消失;例如，Birch和Swinnerton-Dyer猜想指出椭圆曲线$L$ -函数在那里消失的阶等于Mordell-Weil群的秩。Katz和Sarnak密度猜想指出，这种行为和其他行为可以通过随机矩阵集合很好地建模。当检验函数受到适当限制时，这种对应关系对于许多系列都是已知的。对于适当的选择，我们得到的平均阶消失在中心点的家庭。在本说明中，我们报告的进展，在确定最佳的检验函数的各种经典的紧群不同的支持限制，并讨论这与改进的秩界。
英文摘要：
Given an $L$-function, one of the most important questions concerns its vanishing at the central point; for example, the Birch and Swinnerton-Dyer conjecture states that the order of vanishing there of an elliptic curve $L$-function equals the rank of the Mordell-Weil group. The Katz and Sarnak Density Conjecture states that this and other behavior is well-modeled by random matrix ensembles. This correspondence is known for many families when the test functions are suitably restricted. For appropriate choices, we obtain bounds on the average order of vanishing at the central point in families. In this note we report on progress in determining the optimal test functions for the various classical compact groups for different support restrictions, and discuss how this relates to improved rank bounds.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
